Facebook didn't launch 'til February 2004. Initially it was only available at US Ivy League colleges and, then, in 2005 at some international universities, but it was not "unavoidable" in 2004.
Every, if you bothered to get your university internal mail unlocked.
I had to send them a mail with the website of my German university and how our e-mail addresses looked, about 6 weeks later I was able tor register.
Note: I went to a university in the US but not an Ivy